1001.3871v2

arX

iv:1

001.

3871

v2 [

hep-

th]

26

Aug

201

0

DAMTP 201005

A First Course on

Twistors, Integrability and Gluon Scattering

Amplitudes

Martin Wolf

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Abstract

These notes accompany an introductory lecture course on the twistor approach to

supersymmetric gauge theories aimed at early-stage PhD students. It was held by

the author at the University of Cambridge during the Michaelmas term in 2009.

The lectures assume a working knowledge of differential geometry and quantum field

theory. No prior knowledge of twistor theory is required.

21st January 2010

Also at the Wolfson College, Barton Road, Cambridge CB3 9BB, United Kingdom.E-mail address: [email protected]

http://arxiv.org/abs/1001.3871v2mailto:[email protected]

Preface

The course is divided into two main parts: I) The re-formulation of gauge theory on twistor space

and II) the construction of tree-level gauge theory scattering amplitudes. More specifically, the

first few lectures deal with the basics of twistor geometry and its application to free field theories.

We then move on and discuss the non-linear field equations of self-dual YangMills theory. The

subsequent lectures deal with supersymmetric self-dual YangMills theories and the extension to

the full non-self-dual supersymmetric YangMills theory in the case of maximal N = 4 supersym-metry. Whilst studying the field equations of these theories, we shall also discuss the associated

action functionals on twistor space. Having re-interpreted N = 4 supersymmetric YangMillstheory on twistor space, we discuss the construction of tree-level scattering amplitudes. We first

transform, to twistor space, the so-called maximally-helicity-violating amplitudes. Afterwards we

discuss the construction of general tree-level amplitudes by means of the CachazoSvrcekWitten

rules and the BrittoCachazoFengWitten recursion relations. Some mathematical concepts un-

derlying twistor geometry are summarised in several appendices. The computation of scattering

amplitudes beyond tree-level is not covered here.

My main motivation for writing these lecture notes was to provide an opportunity for stu-

dents and researchers in mathematical physics to get a grip of twistor geometry and its ap-

plication to perturbative gauge theory without having to go through the wealth of text books

and research papers but at the same time providing as detailed derivations as possible. Since

the present article should be understood as notes accompanying an introductory lecture course

rather than as an exhaustive review article of the field, I emphasise that even though I tried to

refer to the original literature as accurately as possible, I had to make certain choices for the

clarity of presentation. As a result, the list of references is by no means complete. Moreover,

to keep the notes rather short in length, I had to omit various interesting topics and recent de-

velopments. Therefore, the reader is urged to consult Spires HEP and arXiv.org for the latest

advancements and especially the citations of Wittens paper on twistor string theory, published

in Commun. Math. Phys. 252, 189 (2004), arXiv:hep-th/0312171.

Should you find any typos or mistakes in the text, please let me know by sending an email to

[email protected]. For the most recent version of these lecture notes, please also check

http://www.damtp.cam.ac.uk/user/wolf

Acknowledgements. I am very grateful to J. Bedford, N. Bouatta, D. Correa, N. Dorey,

M. Dunajski, L. Mason, R. Ricci and C. Samann for many helpful discussions and suggestions.

Special thanks go to J. Bedford for various discussions and comments on the manuscript. I would

also like to thank those who attended the lectures for asking various interesting questions. This

work was supported by an STFC Postdoctoral Fellowship and by a Senior Research Fellowship

at the Wolfson College, Cambridge, U.K.

Cambridge, 21st January 2010

Martin Wolf

1

http://www.slac.stanford.edu/spires/hep/search/http://arxiv.org/http://www-spires.dur.ac.uk/cgi-bin/spiface/hep?c=CMPHA,252,189http://www.springerlink.com/content/lxhrcf81x0j73b94/http://arxiv.org/abs/hep-th/0312171mailto:[email protected]://www.damtp.cam.ac.uk/user/wolf

Literature

Amongst many others (see bibliography at the end of this article), the following lecture notes

and books have been used when compiling this article and are recommended as references and

for additional reading (chronologically ordered).

Complex geometry:

(i) P. Griffiths & J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York,

1978

(ii) R. O. Wells, Differential analysis on complex manifolds, Springer Verlag, New York, 1980

(iii) M. Nakahara, Geometry, topology and physics, The Institute of Physics, BristolPhiladel-

phia, 2002

(iv) V. Bouchard, Lectures on complex geometry, CalabiYau manifolds and toric geometry,

arXiv:hep-th/0702063

Supermanifolds and supersymmetry:

(i) Yu. I. Manin, Gauge field theory and complex geometry, Springer Verlag, New York, 1988

(ii) C. Bartocci, U. Bruzzo & D. Hernandez-Ruiperez, The geometry of supermanifolds, Kluwer,

Dordrecht, 1991

(iii) J. Wess & J. Bagger, Supersymmetry and supergravity, Princeton University Press, Prin-

ceton, 1992

(iv) C. Samann, Introduction to supersymmetry, Lecture Notes, Trinity College Dublin, 2009

Twistor geometry:

(i) R. S. Ward & R. O. Wells, Twistor geometry and field theory, Cambridge University Press,

Cambridge, 1989

(ii) S. A. Huggett & K. P. Tod, An introduction to twistor theory, Cambridge University Press,

Cambridge, 1994

(iii) L. J. Mason & N. M. J. Woodhouse, Integrability, self-duality, and twistor theory, Clarendon

Press, Oxford, 1996

(iv) M. Dunajski, Solitons, instantons and twistors, Oxford University Press, Oxford, 2009

Tree-level gauge theory scattering amplitudes and twistor theory:

(i) F. Cachazo & P. Svrcek, Lectures on twistor strings and perturbative YangMills theory,

PoS RTN2005 (2005) 004, arXiv:hep-th/0504194

(ii) J. A. P. Bedford, On perturbative field theory and twistor string theory, arXiv:0709.3478,

PhD thesis, Queen Mary, University of London (2007)

(iii) C. Vergu, Twistors, strings and supersymmetric gauge theories , arXiv:0809.1807, PhD

thesis, Universite Paris IVPierre et Marie Curie (2008)

2

http://arxiv.org/abs/hep-th/0702063http://www.christiansaemann.de/files/LecturesOnSUSY.pdfhttp://arxiv.org/abs/hep-th/0504194http://arxiv.org/abs/0709.3478http://arxiv.org/abs/0809.1807

Contents

Part I: Twistor re-formulation of gauge theory

1. Twistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3. Twistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Massless fields and the Penrose transform . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1. Integral formul for massless fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2. Cech cohomology groups and Penroses theorem a sketch . . . . . . . . . . . . . 12

3. Self-dual YangMills theory and the PenroseWard transform . . . . . . . . . . . . . . . 17

3.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2. PenroseWard transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3. Example: BelavinPolyakovSchwarzTyupkin instanton . . . . . . . . . . . . . . . 23

4. Supertwistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1. A brief introduction to supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2. Supertwistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3. Superconformal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5. Supersymmetric self-dual YangMills theory and the PenroseWard transform . . . . . . 32

5.1. PenroseWard transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2. Holomorphic ChernSimons theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6. N = 4 supersymmetric YangMills theory from supertwistor space . . . . . . . . . . . . 486.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2. N = 4 supersymmetric YangMills theory from supertwistor space . . . . . . . . . 49

Part II: Tree-level gauge theory scattering amplitudes

7. Scattering amplitudes in YangMills theories . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1. Motivation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2. Colour ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.3. Spinor-helicity formalism re-visited . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8. MHV amplitudes and twistor theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.1. Tree-level MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.2. Tree-level MHV superamplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.3. Wittens half Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.4. MHV superamplitudes on supertwistor space . . . . . . . . . . . . . . . . . . . . . 72

9. MHV formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.1. CachazoSvrcekWitten rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.3. MHV diagrams from twistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

9.4. Superamplitudes in the MHV formalism . . . . . . . . . . . . . . . . . . . . . . . . 85

9.5. Localisation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10. BrittoCachazoFengWitten recursion relations . . . . . . . . . . . . . . . . . . . . . . 91

10.1. Recursion relations in pure YangMills theory . . . . . . . . . . . . . . . . . . . . . 91

10.2. Recursion relations in maximally supersymmetric YangMills theory . . . . . . . . 94

3

Appendices

A. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B. Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C. Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D. Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4

Part I

Twistor re-formulation of gauge theory

1. Twistor space

1.1. Motivation

Usually, the equations of motion of physically interesting theories are complicated systems of

coupled non-linear partial differential equations. This thus makes it extremely hard to find explicit

solutions. However, among the theories of interest are some which are completely solvable in the

sense of allowing for the construction (in principle) of all solutions to the corresponding equations

of motion. We shall refer to these systems as integrable systems. It should be noted at this point

that there are various distinct notions of integrability in the literature and here we shall use the

word integrability in the loose sense of complete solvability without any concrete assumptions.

The prime examples of integrable theories are the self-dual YangMills and gravity theories in

four dimensions including their various reductions to lower space-time dimensions. See e.g. [1, 2]

for details.

Twistor theory has turned out to be a very powerful tool in analysing integrable systems. The

key ingredient of twistor theory is the substitution of space-time as a background for physical

processes by an auxiliary space called twistor space. The term twistor space is used collectively

and refers to different spaces being associated with different physical theories under consideration.

All these twistor spaces have one thing in common in that they are (partially) complex manifolds,

and moreover, solutions to the field equations on space-time of the theory in question are encoded

in terms of differentially unconstrained (partially) complex analytic data on twistor space. This

way one may sometimes even classify all solutions to a problem. The goal of the first part of

these lecture notes is the twistor re-formulation of N = 4 supersymmetric YangMills theory onfour-dimensional flat space-time.

1.2. Preliminaries

Let us consider M4 = Rp,q for p+ q = 4, where Rp,q is Rp+q equipped with a metric g = (g) =diag(1p,1q) of signature (p, q). Here and in the following, , , . . . run from 0 to 3. In particular,for (p, q) = (0, 4) we shall speak of Euclidean (E) space, for (p, q) = (1, 3) of Minkowski (M) spaceand for (p, q) = (2, 2) of Kleinian (K) space. The rotation group is then given by SO(p, q). Belowwe shall only be interested in the connected component of the identity of the rotation group

SO(p, q) which is is commonly denoted by SO0(p, q).

If we let , , . . . = 1, 2 and , , . . . = 1, 2, then we may represent any real four-vector

x = (x) M4 as a 2 2-matrix x = (x) Mat(2,C) = C4 subject to the following realityconditions:1 E : x = 2 xt2 ,M : x = xt ,K : x = x , (1.1)

1Note that for the Kleinian case one may alternatively impose x = 1 xt1.

6

where bar denotes complex conjugation, t transposition and i, for i, j, . . . = 1, 2, 3, are the

Pauli matrices

1 =

(

0 1

1 0

)

, 2 =

(

0 ii 0

)

and 3 =

(

1 0

0 1

)

. (1.2)

Recall that they obey

ij = ij + i

k

ijkk , (1.3)

where ij is the Kronecker symbol and ijk is totally anti-symmetric in its indices with 123 = 1.

To be more concrete, the isomorphism : x 7 x = (x) can be written as

x = x x = 12 x , (1.4a)

where = [] with 12 = 1 and = (and similar relations for )2E : ( ) := (12, i3,i2,i1) ,M : ( ) := (i12,i1,i2,i3) ,K : ( ) := (3, 1,i2,12) . (1.4b)The line element ds2 = gdx

dx on M4 = Rp,q is then given byds2 = det dx = 12dx

dx (1.5)

Rotations (respectively, Lorentz transformations) act on x according to x 7 x = x with = () SO0(p, q). The induced action on x reads as

x 7 x = g1 x g2 for g1,2 GL(2,C) . (1.6)The g1,2 are not arbitrary for several reasons. Firstly, any two pairs (g1, g2) and (g

1, g

2) with

(g1, g2) = (tg1, t

1g2) for t C\{0} induce the same transformation on x, hence we may regard theequivalence classes [(g1, g2)] = {(g1, g2)|(g1, g2) = (tg1, t1g2)}. Furthermore, rotations preservethe line element and from det dx = det dx we conclude that det g1 det g2 = 1. Altogether, we

may take g1,2 SL(2,C) without loss of generality. In addition, the g1,2 have to preserve thereality conditions (1.1). For instance, on E we find that g1,2 = 1 g1,2 t1. Explicitly, we have

g1,2 =

(

a1,2 b1,2

c1,2 d1,2

)

=

(

a1,2 b1,2

b1,2 a1,2

)

. (1.7)

Since det g1,2 = 1 = |a1,2|2 + |b1,2|2 (which topologically describes a three-sphere) we concludethat g1,2 SU(2), i.e. g11,2 = g

1,2. In addition, if g1,2 SU(2) then also g1,2 SU(2) and since

g1,2 and g1,2 induce the same transformation on x, we have therefore established

SO(4) = (SU(2) SU(2))/Z2 . (1.8)2We have chosen particle physics literature conventions which are somewhat different from the twistor literature.

7

One may proceed similarly for M and K but we leave this as an exercise. Eventually, wearrive atE : SO(4) = (SU(2) SU(2))/Z2 , with x 7 g1 x g2 and g1,2 SU(2) ,M : SO0(1, 3) = SL(2,C)/Z2 , with x 7 g x g and g SL(2,C) ,K : SO0(2, 2) = (SL(2,R) SL(2,R))/Z2 , with x 7 g1 x g2 and g1,2 SL(2,R) .

(1.9)

Notice that in general one may write

SO0(p, q) = Spin(p, q)/Z2 , (1.10)where Spin(p, q) is known as the spin group of Rp,q. In a more mathematical terminology,Spin(p, q) is the double cover of SO0(p, q) (for the sum p + q not necessarily restricted to 4).

For p = 0, 1 and q > 2, the spin group is simply connected and thus coincides with the univer-

sal cover. Since the fundamental group (or first homotopy group) of Spin(2, 2) is non-vanishing,

1(Spin(2, 2)) = ZZ, the spin group Spin(2, 2) is not simply connected. See, e.g. [3,4] for moredetails on the spin groups.

In summary, we may either work with x or with x and making this identification amounts

to identifying g with12. Different signatures are encoded in different reality conditions

(1.1) on x. Hence, in the following we shall work with the complexification M4 C = C4 andx = (x) Mat(2,C) and impose the reality conditions whenever appropriate. Therefore, thedifferent cases of (1.9) can be understood as different real forms of the complex version

SO(4,C) = (SL(2,C) SL(2,C))/Z2 . (1.11)For brevity, we denote x by x and M4 C by M4.

Exercise 1.1. Prove that the rotation groups on M and K are given by (1.9).1.3. Twistor space

In this section, we shall introduce Penroses twistor space [5] by starting from complex space-time

M4 = C4 and the identification x x . According to the discussion of the previous section,we view the tangent bundle TM4 of M4 according to

TM4 = S S ,

:=

x :=

x,

(1.12)

where S and S are the two complex rank-2 vector bundles called the bundles of dotted and

undotted spinors. See Appendix A. for the definition of a vector bundle. The two copies of

SL(2,C) in (1.11) act independently on S and S. Let us denote undotted spinors by anddotted ones by .3 On S and S we have the symplectic forms and from before which

3Notice that it is also common to denote undotted spinors by and dotted spinors by . However, we shall

stick to our above conventions.

8

can be used to raise and lower spinor indices:

= and =

. (1.13)

Remark 1.1. Let us comment on conformal structures since the identification (1.12)

amounts to choosing a (holomorphic) conformal structure. This can be seen as follows:

The standard definition of a conformal structure on a four-dimensional complex manifold

X states that a conformal structure is an equivalence class [g], the conformal class, of holo-

morphic metrics g on X, where two given metrics g and g are called equivalent if g = 2g

for some nowhere vanishing holomorphic function . Put differently, a conformal structure

is a line subbundle L in T X T X. Another, maybe less familiar definition assumes afactorisation of the holomorphic tangent bundle TX of X as a tensor product of two rank-2

holomorphic vector bundles S and S, that is, TX = S S. This isomorphism in turn gives(canonically) the line subbundle 2S2S in T XT X which, in fact, can be identifiedwith L. The metric g is then given by the tensor product of the two symplectic forms on S

and S (as done above) which are sections of 2S and 2S.

Let us now consider the projectivisation of the dual spin bundle S. Since S is of rank two,

the projectivisation P(S) M4 is a CP 1-bundle over M4. Hence, P(S) is a five-dimensionalcomplex manifold bi-holomorphic to C4 CP 1. In what follows, we shall denote it by F 5 andcall it correspondence space. The reason for this name becomes transparent momentarily. We

take (x , ) as coordinates on F5, where are homogeneous coordinates on CP 1.

Remark 1.2. Remember that CP 1 can be covered by two coordinate patches, U, withCP 1 = U+ U. If we let = (1, 2)t be homogeneous coordinates on CP 1 with t for t C \ {0}, U and the corresponding affine coordinates can be defined asfollows:

U+ : 1 6= 0 and + :=21

,

U : 2 6= 0 and :=12

.

On U+ U = C \ {0} we have + = 1 .On F 5 we may consider the following vector fields:

V = =

x. (1.14)

They define an integrable rank-2 distribution on F 5 (i.e. a rank-2 subbundle in TF 5) which is

called the twistor distribution. Therefore, we have a foliation of F 5 by two-dimensional complex

manifolds. The resulting quotient will be twistor space, a three-dimensional complex manifold

9

denoted by P 3. We have thus established the following double fibration:

P 3 M4

F 5

1 2

@@R

(1.15)

The projection 2 is the trivial projection and 1 : (x , ) 7 (z, ) = (x, ), where

(z, ) are homogeneous coordinates on P3. The relation

z = x (1.16)

is known as the incidence relation. Notice that (1.15) makes clear why F 5 is called correspondence

space: It is the space that links space-time with twistor space.

Also P 3 can be covered by two coordinate patches, which we (again) denote by U (see also

Remark 1.2.):

U+ : 1 6= 0 and z+ :=z

1and + :=

21

,

U : 2 6= 0 and z :=z

2and :=

12

.

(1.17)

On U+ U we have z+ = +z and + = 1 . This shows that twistor space P 3 can beidentified with the total space of the holomorphic fibration

O(1)O(1) CP 1 , (1.18)where O(1) is the dual of the tautological line bundle O(1) over CP 1,

O(1) := {(, ) CP 1 C2 | } , (1.19)i.e. O(1) = O(1). The bundle O(1) is also referred to as the hyperplane bundle. Other linebundles, which we will frequently encounter below, are:

O(m) = O(1)m and O(m) = O(m) for m N . (1.20)The incidence relation z = x identifies x M4 with holomorphic sections of (1.18). Notethat P 3 can also be identified with CP 3 \ CP 1, where the deleted projective line is given byz 6= 0 and = 0.

Exercise 1.2. Let be homogeneous coordinates on CP 1 and z be the fibre coordinatesof O(m) CP 1 for m Z. Furthermore, let {U} be the canonical cover as in Remark1.2. Show that the transition function of O(m) is given by m+ = m . Show further thatwhile O(1) has global holomorphic sections, O(1) does not.

Having established the double fibration (1.15), we may ask about the geometric correspond-

ence, also known as the Klein correspondence, between space-time M4 and twistor space P 3. In

fact, for any point x M4, the corresponding manifold Lx := 1(12 (x)) P 3 is a curve which

10

is bi-holomorphic to CP 1. Conversely, any point p P 3 corresponds to a totally null-plane inM4, which can be seen as follows. For some fixed p = (z, ) P 3, the incidence relation (1.16)tells us that x = x0 +

since = = 0. Here, x0 is a particular solution to

(1.16). Hence, this describes a two-plane in M4 which is totally null since any null-vector x is of

the form x = . In addition, (1.16) implies that the removed line CP 1 of P 3 = CP 3 \CP 1corresponds to the point infinity of space-time. Thus, CP 3 can be understood as the twistorspace of conformally compactified complexified space-time.

Remark 1.3. Recall that a four-vector x in M4 is said to be null if it has zero norm, i.e.

gxx = 0. This is equivalent to saying that detx = 0. Hence, the two columns/rows of

x must be linearly dependent. Thus, x = .

2. Massless fields and the Penrose transform

The subject of this section is to sketch how twistor space can be used to derive all solutions to

zero-rest-mass field equations.

2.1. Integral formul for massless fields

To begin with, let P 3 be twistor space (as before) and consider a function f that is holomorphic

on the intersection U+ U P 3. Furthermore, let us pull back f to the correspondence spaceF 5. The pull-back of f(z, ) is f(x

, ), since the tangent spaces of the leaves of the

fibration 1 : F5 P 3 are spanned by (1.14) and so the pull-backs have to be annihilated by

the vector fields (1.14). Then we may consider following contour integral:

(x) = 12i

C

d f(x , ) , (2.1)

where C is a closed curve in U+U CP 1.4 Since the measure d is of homogeneity 2, thefunction f should be of homogeneity 2 as only then is the integral well-defined. Put differently,only if f is of homogeneity 2, is a function defined on M4.

Furthermore, one readily computes

= 0 , with := 12 (2.2)

by differentiating under the integral. Hence, the function satisfies the KleinGordon equation.

Therefore, any f with the above properties will yield a solution to the KleinGordon equation via

the contour integral (2.1). This is the essence of twistor theory: Differentially constrained data

on space-time (in the present situation the function ) is encoded in differentially unconstrained

complex analytic data on twistor space (in the present situtation the function f).

4As before, we shall not make any notational distinction between the coordinate patches covering CP 1 and theones covering twistor space.

11

Exercise 2.1. Consider the following function f = 1/(z1z2) which is holomorphic on U+U P 3. Clearly, it is of homogeneity 2. Show that the integral (2.1) gives rise to = 1/det x. Hence, this f yields the elementary solution to the KleinGordon equation

based at the origin x = 0.

What about the other zero-rest-mass field equations? Can we say something similar about

them? Consider a zero-rest-mass field 12h of positive helicity h (with h > 0). Then

12h(x) = 1

2i

C

d 1 2hf(x, ) (2.3)

solves the equation

112h = 0 . (2.4)

Again, in order to have a well-defined integral, the integrand should have total homogeneity zero,

which is equivalent to requiring f to be of homogeneity 2h 2. Likewise, we may also considera zero-rest-mass field 12h of negative helicity h (with h > 0) for which we take

12h(x) = 1

2i

C

d

z1

z2hf(x, ) (2.5)

such that f is of homogeneity 2h 2. Hence,

112h = 0 . (2.6)

These contour integral formul provide the advertised Penrose transform [6, 7]. Sometimes, one

refers to this transform as the RadonPenrose transform to emphasise that it is a generalisation

of the Radon transform.5

In summary, any function on twistor space, provided it is of appropriate homogeneity m Z,can be used to construct solutions to zero-rest-mass field equations. However, there are a lot of

different functions leading to the same solution. For instance, we could simply change f by adding

a function which has singularities on one side of the contour but is holomorphic on the other,

since the contour integral does not feel such functions. How can we understand what is going

on? Furthermore, are the integral formul invertible? In addition, we made use of particular

coverings, so do the results depend on these choices? The tool which helps clarify all these issues

is sheaf cohomology.6 For a detailed discussion about sheaf theory, see e.g. [4, 9].

2.2. Cech cohomology groups and Penroses theorem a sketch

Consider some Abelian sheaf S over some manifold X, that is, for any open subset U X onehas an Abelian group S(U) subject to certain locality conditions; Appendix D. collects useful

5 The Radon transform, named after Johann Radon [8], is an integral transform in two dimensions consisting

of the integral of a function over straight lines. It plays an important role in computer assisted tomography. The

higher dimensional analog of the Radon transform is the X-ray transform; see footnote 26.6In Section 8.3. we present a discussion for Kleinian signature which by-passes sheaf cohomology.

12

definitions regarding sheaves including some examples. Furthermore, let U = {Ui} be an opencover of X. A q-cochain of the covering U with values in S is a collection f = {fi0iq} of sectionsof the sheaf S over non-empty intersections Ui0 Uiq .

The set of all q-cochains has an Abelian group structure (with respect to addition) and is

denoted by Cq(U,S). Then we define the coboundary map by

q : Cq(U,S) Cq+1(U,S) ,

(qf)i0iq+1 :=q+1

k=0

()iri0ikiq+1i0iq+1 fi0ik iq+1 ,(2.7a)

where

ri0ikiq+1i0iq+1 : S(Ui0 Uik Uiq+1) S(Ui0 Uiq+1) (2.7b)

is the sheaf restriction morphism and ik means omitting ik. It is clear that q is a morphism of

groups, and one may check that q q1 = 0.

Exercise 2.2. Show that q q1 = 0 for q as defined above.

Furthermore, we see straight away that ker 0 = S(X). Next we define

Zq(U,S) := ker q and Bq(U,S) := im q1 . (2.8)

We call elements of Zq(U,S) q-cocycles and elements of Bq(U,S) q-coboundaries, respectively.Cocycles are anti-symmetric in their indices. Both Zq(U,S) and Bq(U,S) are Abelian groups andsince the coboundary map is nil-quadratic, Bq(U,S) is a (normal) subgroup of Zq(U,S). Theq-th Cech cohomology group is the quotient

Hq(U,S) := Zq(U,S)/Bq(U,S) . (2.9)

In order to get used to these definitions, let us consider a simple example and take the

(Abelian) sheaf of holomorphic sections of the line bundle O(m) CP 1. As before we choosethe canonical cover U = {U} of CP 1. Since there is only a double intersection, all cohomologygroups Hq with q > 1 vanish automatically. The following table then summarises H0 and H1:

m 4 3 2 1 0 1 2 H0(U,O(m)) 0 0 0 0 C1 C2 C3 H1(U,O(m)) C3 C2 C1 0 0 0 0

Table 2.1: Cech cohomology groups for O(m) CP 1 with respect to the cover U = {U}.Note that when writing Hq(X,E) for some vector bundle E X over some manifold X, weactually mean the (Abelian) sheaf E of sections (either smooth or holomorphic depending on thecontext) of E. By a slight abuse of notation, we shall often not make a notational distinction

between E and its sheaf of sections E and simply write E in both cases.

13

Let us now compute H1(U,O(m)) for m 0. The rest is left as an exercise. To this end,consider some representative f = {f+} defined on U+ U CP 1.7 Clearly, 1f = 0 as thereare no triple intersections. Without loss of generality, f might be taken as

f+ =1

(1)m

n=cn

(21

)n

. (2.10)

This can be re-written according to

f+ =1

(1)m

n=cn

(21

)n

=1

(1)m

[ m

n=+

1

n=m+1+

n=0

]

cn

(21

)n

=1

(1)m

n=0

cn

(21

)n

=: r++f+

+m1

n=1

cn(2)

n(1)mn

=: f +

+1

(2)m

n=0

cnm

(12

)n

=: r+f

= f + + r++f+ r+f , (2.11)

where r+ are the restriction mappings. Since the f are holomorphic on U, we conclude that

f = {f+} is cohomologous to f = {f +} with

f + =m1

n=1

cn(2)

n(1)mn . (2.12)

There are precisely m 1 independent complex parameters, c1, . . . , cm+1, which parametrisef . Hence, we have established H1(U,O(m)) = Cm1 whenever m > 1 and H1(U,O(m)) = 0for m = 0, 1.

Exercise 2.3. Complete the Table 2.1.

Table 2.1. hints that there is some sort of duality. In fact,

H0(U,O(m)) = H1(U,O(m 2)) , (2.13)

which is a special instance of Serre duality (see also Remark 2.1.). Here, the star denotes the

vector space dual. To understand this relation better, consider (m 0)

g H0(U,O(m)) , with g = g1m1 m (2.14)

and f H1(U,O(m 2)). Then define the pairing

(f, g) := 12i

C

d f() g() , (2.15)

7Notice that in the preceding sections, we have not made a notational distinction between f and f+, but

strictly speaking we should have.

14

where the contour is chosen as before. This expression is complex linear and non-degenerate and

depends only on the cohomology class of f . Hence, it gives the duality (2.13).

A nice way of writing (2.15) is as (f, g) = f1mg1m , where

f1m := 1

2i

C

d 1 m f() , (2.16)

such that Penroses contour integral formula (2.1) can be recognised as an instance of Serre duality

(the coordinate x being interpreted as some parameter).

Remark 2.1. If S is some Abelian sheaf over some compact complex manifold X withcovering U and K the sheaf of sections of the canonical line bundle K := detT X, thenthere is the following isomorphism which is referred to as Serre duality (or sometimes to as

KodairaSerre duality):

Hq(U,S) = Hdq(U,S K) .

Here, d = dimCX. See e.g. [9] for more details. In our present case, X = CP 1 and sod = 1 and K = detT CP 1 = T CP 1 = O(2) and furthermore S = O(m).One technical issue remains to be clarified. Apparently all of our above calculations seem to

depend on the chosen cover. But is this really the case?

Consider again some manifold X with cover U together with some Abelian sheaf S. If anothercover V is the refinement of U, that is, for U = {Ui}iI and V = {Vj}jJ there is a map : J Iof index sets, such that for any j J , Vj U(j), then there is a natural group homomorphism(induced by the restriction mappings of the sheaf S)

hUV : Hq(U,S) Hq(V,S) . (2.17)

We can then define the inductive limit of these cohomology groups with respect to the partially

ordered set of all coverings (see also Remark 2.2.),

Hq(X,S) := lim indU

Hq(U,S) (2.18)

which we call the q-th Cech cohomology group of X with coefficients in S.

Remark 2.2. Let us recall the definition of the inductive limit. If we let I be a partially

ordered set (with respect to ) and Si a family of modules indexed by I with homomorph-isms f ij : Si Sj with i j and f ii = id, f ij f jk = f ik for i j k, then the inductivelimit,

lim indiI

Si ,

is defined by quotienting the disjoint union

iISi =

iI{(i, Si)} by the following equival-ence relation: Two elements xi and xj of

iISi are said to be equivalent if there exists a

k I such that f ik(xi) = fjk(xj).

15

By the properties of inductive limits, we have a homomorphism Hq(U,S) Hq(X,S). Now thequestion is: When does this becomes an isomorphism? The following theorem tells us when this

is going to happen.

Theorem 2.1. (Leray) Let U = {Ui} be a covering of X with the property that for all tuples(Ui0 , . . . , Uip) of the cover, H

q(Ui0 Uip ,S) = 0 for all q 1. Then

Hq(U,S) = Hq(X,S) .

For a proof, see e.g. [10, 9].

Such covers are called Leray or acyclic covers and in fact our two-set cover U = {U} of CP 1is of this form. Therefore, Table 2.1. translates into Table 2.2.

m 4 3 2 1 0 1 2 H0(CP 1,O(m)) 0 0 0 0 C1 C2 C3 H1(CP 1,O(m)) C3 C2 C1 0 0 0 0

Table 2.2: Cech cohomology groups for O(m) CP 1.Remark 2.3. We have seen that twistor space P 3 = CP 3 \CP 1 = O(1)O(1); see (1.17)and (1.18). There is yet another interpretation. The Riemann sphere CP 1 can be embeddedinto CP 3. The normal bundle NCP 1|CP 3 of CP 1 inside CP 3 is O(1)O(1) as follows fromthe short exact sequence:

01 TCP 1 2 TCP 3|CP 1 3 NCP 1|CP 3 4 0 .

Exactness of this sequence means that imi = keri+1. If we take (z, ) as homogeneous

coordinates on CP 3 with the embedded CP 1 corresponding to z = 0 and 6= 0, thenthe non-trivial mappings 2,3 are given by 2 : / 7 / while 3 : /z +/ 7 , where , are linear in z, and the restriction to CP 1 is understood.This shows that indeed NCP 1|CP 3 = O(1) O(1), i.e. twistor space P 3 can be identifiedwith the normal bundle of CP 1 CP 3. Kodairas theorem on relative deformation statesthat if Y is a compact complex submanifold of a not necessarily compact complex manifold

X, and if H1(Y,NY |X) = 0, where NY |X is the normal bundle of Y in X, then there exists

a d-dimensional family of deformations of Y inside X, where d := dimCH0(Y,NY |X). Seee.g. [11, 12] for more details. In our example, Y = CP 1, X = CP 3 and NCP 1|CP 3 =O(1) O(1). Using Table 2.2., we conclude that H1(CP 1,O(1) O(1)) = 0 and d = 4.In fact, complex space-time M4 = C4 is precisely this family of deformations. To be moreconcrete, any Lx = CP 1 has O(1)C2 as normal bundle, and the tangent space TxM4 atx M4 arises as TxM4 = H0(Lx,O(1) C2) = H0(Lx,C2) H0(Lx,O(1)) = Sx Sx,where Sx := H

0(Lx,C2) and Sx := H0(Lx,O(1)) which is the factorisation (1.12).16

In summary, the functions f on twistor space from Section 2.1. leading to solutions of zero-

rest-mass field equations should be thought of as representatives of sheaf cohomology classes in

H1(P 3,O(2h 2)). Then we can state the following theorem:

Theorem 2.2. (Penrose [7]) If we let Zh be the sheaf of (sufficiently well-behaved) solutions tothe helicity h (with h 0) zero-rest-mass field equations on M4, then

H1(P 3,O(2h 2)) = H0(M4,Zh) .

The proof of this theorem requires more work including a weightier mathematical machinery. It

therefore lies somewhat far afield from the main thread of development and we refer the interested

reader to e.g. [4] for details.

3. Self-dual YangMills theory and the PenroseWard transform

So far, we have discussed free field equations. The subject of this section is a generalisation

of our above discussion to the non-linear field equations of self-dual YangMills theory on four-

dimensional space-time. Selfdual YangMills theory can be regarded as a subsector of Yang

Mills theory and in fact, the selfdual YangMills equations are the Bogomolnyi equations of

YangMills theory. Solutions to the self-dual YangMills equations are always solutions to the

YangMills equations, while the converse may not be true.

3.1. Motivation

To begin with, let M4 be E and E M4 a (complex) vector bundle over M4 with structuregroup G. For the moment, we shall assume that G is semi-simple and compact. This allows us

to normalise the generators ta of G according to tr(tatb) = tr(tatb) = C(r)ab with C(r) > 0.

Furthermore, let : p(M4, E) p+1(M4, E) be a connection on E with curvature F = 2 H0(M4,2(M4,EndE)). Here, p(M4) are the p-forms on M4 and p(M4, E) := p(M4) E.Then = d + A and F = dA+ A A, where A is the EndE-valued connection one-form. Thereader unfamiliar with these quantities may wish to consult Appendix A. for their definitions. In

the coordinates x on M4 we have

A = dxA and = dx , with = +A (3.1a)

and therefore

F = 12dx dxF , with F = [, ] = A A + [A, A ] . (3.1b)

The YangMills action functional is defined by

S = 1g2YM

M4tr(F F ) , (3.2)

where gYM is the YangMills coupling constant and denotes the Hodge star on M4. Thecorresponding field equations read as

F = 0 F = 0 . (3.3)

17

Exercise 3.1. Derive (3.3) by varying (3.2).

Solutions to the YangMills equations are critical points of the YangMills action. The critical

points may be local maxima of the action, local minima, or saddle points. To find the field

configurations that truly minimise (3.2), we consider the following inequality:

M4tr[(F F ) (F F )

] 0 . (3.4)

A short calculation then shows that

M4tr(F F )

M4tr(F F ) (3.5)

and therefore

S 1g2YM

M4tr(F F ) = S 8

2

g2YM|Q| , (3.6)

where Q Z is called topological charge or instanton number,Q = 1

82

M4tr(F F ) = c2(E) . (3.7)

Here, c2(E) denotes the second Chern class of E; see Appendix A. for the definition.

Equality is achieved for configurations that obey

F = F F = 12F (3.8)

with = [] and 0123 = 1. These equations are called the self-dual and anti-self-

dual YangMills equations. Solutions to these equations with finite charge Q are referred to as

instantons and anti-instantons. The sign of Q has been chosen such that Q > 0 for instantons

while Q < 0 for anti-instantons. Furthermore, by virtue of the Bianchi identity, F = 0 [F] = 0, solutions to (3.8) automatically satisfy the second-order YangMills equations (3.3).

Remember from our discussion in Section 1.2. that the rotation group SO(4) is given by

SO(4) = (SU(2) SU(2))/Z2 . (3.9)Therefore, the anti-symmetric tensor product of two vector representations 4 4 decomposesunder this isomorphism as 4 4 = 3 3. More concretely, by taking the explicit isomorphism(1.4), we can write

F :=14

F = f + f , (3.10)

with f = f and f = f. Since each of these symmetric rank-2 tensors has three inde-

pendent components, we have made the decomposition 4 4 = 3 3 explicit. Furthermore, ifwe write F = F+ + F with F := 12(F F ), i.e. F = F, then

F+ f and F f . (3.11)

Therefore, the self-dual YangMills equations correspond to

F = F F = 0 f = 0 (3.12)

and similarly for the anti-self-dual YangMills equations.

18

Exercise 3.2. Verify (3.10) and (3.11) explicitly. Show further that F F correspondsto ff

+ ff while F F to ff ff .

Most surprisingly, even though they are non-linear, the (anti-)self-dual YangMills equations

are integrable in the sense that one can give, at least in principle, all solutions. We shall establish

this by means of twistor geometry shortly, but again we will not be too rigorous in our discussion.

Furthermore, f = 0 or f = 0 make perfect sense in the complex setting. For convenience,

we shall therefore work in the complex setting from now on and impose reality conditions later

on when necessary. Notice that contrary to the Euclidean and Kleinian cases, the (anti-)self-dual

YangMills equations on Minkowski space only make sense for complex Lie groups G. This is so

because 2 = 1 on two-forms in Minkowski space.

3.2. PenroseWard transform

The starting point is the double fibration (1.15), which we state again for the readers convenience,

P 3 M4

F 5

1 2

@@R

(3.13)

Consider now a rank-r holomorphic vector bundle E P 3 together with its pull-back 1E F 5.Their structure groups are thus GL(r,C). We may impose the additional condition of having atrivial determinant line bundle, detE, which reduces GL(r,C) to SL(r,C). Furthermore, weagain choose the two-patch covering U = {U} of P 3. Similarly, F 5 may be covered by twocoordinate patches which we denote by U = {U}. Therefore, E and 1E are characterised bythe transition functions f = {f+} and 1f = {1f+}. As before, the pull-back of f+(z, )is f+(x, ), i.e. it is annihilated by the vector fields (1.14) and therefore constant along

1 : F5 P 3. In the following, we shall not make a notational distinction between f and 1f

and simply write f for both bundles. Letting P and F be the anti-holomorphic parts of the

exterior derivatives on P 3 and F 5, respectively, we have 1 P = F 1 . Hence, the transitionfunction f+ is also annihilated by F .

We shall also assume that E is holomorphically trivial when restricted to any projective line

Lx = 1(12 (x)) P 3 for x M4. This then implies that there exist matrix-valued functions

on U, which define trivialisations of 1E over U , such that f+ can be decomposed as (see

also Remark 3.1.)

f+ = 1+ (3.14)

with F = 0, i.e. the = (x, ) are holomorphic on U. Clearly, this splitting is not

unique, since one can always perform the transformation

7 g1 , (3.15)

19

where g is some globally defined matrix-valued function holomorphic function on F 5. Hence, it

is constant on CP 1, i.e. it depends on x but not on . We shall see momentarily, what thetransformation (3.15) corresponds to on space-time M4.

Since V f+ = 0, where V are the restrictions of the vector fields V given in (1.14) to the

coordinate patches U, we find

+V+

1+ = V

+

1 (3.16)

on U+ U. Explicitly, V = with + := /1 = (1, +)t and := /2 = (, 1)t.Therefore, by an extension of Liouvilles theorem, the expressions (3.16) can be at most linear in

+. This can also be understood by noting

+V+

1+ = V

+

1 = +V

1 (3.17)

and so it is of homogeneity 1. Thus, we may introduce a Lie algebra-valued one-form A on F 5

which has components only along 1 : F5 P 3,

VyA|U := A = V

1 =

A , (3.18)

where A is -independent. This can be re-written as

(V +A ) =

= 0 , with := +A . (3.19)

The compatibility conditions for this linear system read as

[,] + [ ,] = 0 , (3.20)

which is equivalent to saying that the f-part of

[,] = f + f (3.21)

vanishes. However, f = 0 is nothing but the self-dual YangMills equations (3.12) on M4.

Notice that the transformations of the form (3.15) induce the transformations

A 7 g1g + g1Ag (3.22)

of A as can be seen directly from (3.19). Hence, these transformations induce gauge trans-

formations on space-time and so we may define gauge equivalence classes [A ], where two gauge

potentials are said to be equivalent if they differ by a transformation of the form (3.22). On the

other hand, transformations of the form8

f+ 7 h1+ f+h , (3.23)

where h are matrix-valued functions holomorphic on U with V h = 0, leave the gauge

potential A invariant. Since V h = 0, the functions h descend down to twistor space P

3

8In Section 5.2., we will formalise these transformations in the framework of non-Abelian sheaf cohomology.

20

and are holomorphic on U (remember that any function on twistor space that is pulled back to

the correspondence space must be annihilated by the vector fields V). Two transition functions

that differ by a transformation of the form (3.23) are then said to be equivalent, as they define

two holomorphic vector bundles which are bi-holomorphic. Therefore, we may conclude that an

equivalence class [f+] corresponds to an equivalence class [A ].

Altogether, we have seen that holomorphic vector bundles E P 3 over twistor space, whichare holomorphically trivial on all projective lines Lx P 3 yield solutions of the self-dual YangMills equations on M4. In fact, the converse is also true: Any solution to the self-dual YangMills

equations arises in this way. See e.g. [4] for a complete proof. Therefore, we have:

Theorem 3.1. (Ward [13]) There is a one-to-one correspondence between gauge equivalence

classes of solutions to the self-dual YangMills equations on M4 and equivalence classes of holo-

morphic vector bundles over the twistor space P 3 which are holomorphically trivial on any pro-

jective line Lx = 1(12 (x)) P 3.

Hence, all solutions to the self-dual YangMills equations are encoded in these vector bundles and

once more, differentially constrained data on space-time (the gauge potential A) is encoded in

differentially unconstrained complex analytic data (the transition function f+) on twistor space.

The reader might be worried that our constructions depend on the choice of coverings, but as in

the case of the Penrose transform, this is not the case as will become transparent in Section 5.2.

As before, one may also write down certain integral formul for the gauge potential A . In

addition, given a solution A = dxA to the self-dual YangMills equations, the matrix-valued

functions are given by

= Pexp(

C

A

)

, (3.24)

where P denotes the path-ordering symbol and the contour C is any real curve in the null-plane

2(11 (p)) M4 for p P 3 running from some point x0 to a point x with x(s) = x0 +s

for s [0, 1] and constant ; the choice of contour plays no role, since the curvature is zero whenrestricted to the null-plane.

Exercise 3.3. Show that for a rank-1 holomorphic vector bundle E P 3, the Ward the-orem coincides with the Penrose transform for a helicity h = 1 field. See also Appendix D.Thus, the Ward theorem gives a non-Abelian generalisation of that case and one therefore

often speaks of the PenroseWard transform.

Before giving an explicit example of a real instanton solution, let us say a few words about

real structures. In Section 1.2., we introduced reality conditions on M4 leading to Euclidean,

Minkowski and Kleinian spaces. In fact, these conditions are induced from twistor space as we

shall now explain. For concreteness, let us restrict our attention to the Euclidean case. The

Kleinian case will be discussed in Section 8.3. Remember that a Minkowski signature does not

allow for real (anti-)instantons.

21

A real structure on P 3 is an anti-linear involution : P 3 P 3. We may choose it accordingto:

(z, ) := (zC

, C) , (3.25a)

where bar denotes complex conjugation as before and

(C) :=

(

0 1

1 0

)

and (C) :=

(

0 11 0

)

. (3.25b)

By virtue of the incidence relation z = x, we obtain an induced involution on M4,9

(x) = xCC . (3.26)

The set of fixed points (x) = x is given by x11 = x22 and x12 = x12. By inspecting (1.1), wesee that this corresponds to a Euclidean signature real slice E in M4. Furthermore, can beextended to E P 3 according to f+(z, ) = (f+((z, )).10 This will ensure that the YangMills gauge potential on space-time is real and in particular, we find from (3.19) that A = A.Here, denotes Hermitian conjugation.

Remark 3.1. Let us briefly comment on generic holomorphic vector bundles over CP 1:So, let E CP 1 be a rank-r holomorphic vector bundle over CP 1. The BirkhoffGrothendieck theorem (see e.g. [9] for details) then tells us that E always decomposes into

a sum of holomorphic line bundles,

O(k1) O(kr) CP 1 .Therefore, if U = {U} denotes the canonical cover of CP 1, the transition function f ={f+} of E is always of the form

f+ = 1+ + , with + := diag(

k1+ , . . . ,

kr+ ) ,

where the are holomorphic on U. If detE is trivial then

i ki = 0. If furthermore E

is holomorpically trivial then ki = 0 and f+ = 1+ .

Notice that given some matrix-valued function f+ which is holomorphic on U+ U CP 1, the problem of trying to split f+ according to f+ = 1+ with holomorphicon U is known as the RiemannHilbert problem and its solutions define holomorphically

trivial vector bundles on CP 1. If in addition f+ also depends on some parameter (in ourabove case the parameter is x), then one speaks of a parametric RiemannHilbert problem.

A solution to the parametric RiemannHilbert problem might not exist for all values of

the parameter, but if it exists at some point in parameter space, then it exists in an open

neighbourhood of that point.

9We shall use the same notation for the anti-holomorphic involutions induced on the different manifolds

appearing in (3.13).10In fact, the involution can be extended to any holomorphic function.

22

3.3. Example: BelavinPolyakovSchwarzTyupkin instanton

Let us now present an explicit instanton solution on Euclidean space for the gauge group SU(2).

This amounts to considering a rank-2 holomorphic vector bundle E P 3 holomorphically trivialon any Lx P 3 with trivial determinant line bundle detE and to equipping twistor space withthe real structure according to our previous discussion.

Then let E P 3 and 1E F 5 be defined by the following transition function f = {f+}[14]:

f+ =1

2

2 z1z212

(z2)2

12

(z1)212 2 + z

1z2

12

, (3.27)

where R \ {0}. Evidently, det f+ = 1 and so detE is trivial. Furthermore, f+(z, ) =(f+((z, )), where is the involution (3.25) leading to Euclidean space. The main problem

now is to find a solution to the RiemannHilbert problem f+ = 1+ . Notice that if we

succeed, we have automatically shown that E P 3 is holomorphically trivial on any projectiveline Lx P 3.

In terms of the coordinates on U+, we have

f+ =1

2

2 z

1+z

2+

+

(z2+)2

+

(z1+)

2

+2 +

z1+z2+

+

. (3.28)

As there is no generic algorithm, let us just present a solution [14]:

+ = 1

1x2 + 2

(

x22z1+ + 2 x22z2+

x12z1+ x12z2+ + 2

)

and = +f+ , (3.29)

where x2 := detx.

It remains to determine the gauge potential and the curvature. We find

A11 =1

2(x2 + 2)

(

x22 0

2x12 x22

)

, A21 =1

2(x2 + 2)

(

x12 2x22

0 x12

)

(3.30)

and A2 = 0. Hence, our choice of gauge 7 g1 corresponds to gauging away A2.Furthermore, the only non-vanishing components of the curvature are

f11 =22

(x2 +2)2

(

0 0

1 0

)

, f12 =2

(x2 +2)2

(

1 0

0 1

)

,

f22 = 22

(x2 + 2)2

(

0 1

0 0

)

,

(3.31)

which shows that we have indeed found a solution to the self-dual YangMills equations. Finally,

using (3.7), we find that the instanton charge Q = 1. We leave all the details as an exercise. The

above solution is the famous BelavinPolyakovSchwarzTyupkin instanton [15]. Notice that

is referred to as the size modulus as it determines the size of the instanton. In addition,

there are four translational moduli corresponding to shifts of the form x 7 x+ c for constant c.Altogether, there are five moduli characterising the charge one SU(2) instanton. For details on

how to construct general instantons, see e.g. [16, 17].

23

Exercise 3.4. Show that (3.29) implies (3.30) and (3.31) by using the linear system (3.19).

Furthermore, show that Q = 1. You might find the following integral useful:

E d4x(x2 + 2)4 = 264 ,where x2 = xx

.

4. Supertwistor space

Up to now, we have discussed the purely bosonic setup. As our goal is the construction of amp-

litudes in supersymmetric gauge theories, we need to incorporate fermionic degrees of freedom.

To this end, we start by briefly discussing supermanifolds before we move on and introduce su-

pertwistor space and the supersymmetric generalisation of the self-dual YangMills equations.

For a detailed discussion about supermanifolds, we refer to [1820].

4.1. A brief introduction to supermanifolds

Let R = R0 R1 be a Z2-graded ring, that is, R0R0 R0, R1R0 R1, R0R1 R1 andR1R1 R0. We call elements of R0 Gramann even (or bosonic) and elements of R1 Gramannodd (or fermionic). An element of R is said to be homogeneous if it is either bosonic or fermionic.

The degree (or Gramann parity) of a homogeneous element is defined to be 0 if it is bosonic and

1 if it is fermionic, respectively. We denote the degree of a homogeneous element r R by pr (pfor parity).

We define the supercommutator, [, } : RR R, by

[r1, r2} := r1r2 ()pr1pr2 r2r1 , (4.1)

for all homogeneous elements r1,2 R. The Z2-graded ring R is called supercommutative if thesupercommutator vanishes for all of the rings elements. For our purposes, the most important

example of such a supercommutative ring is the Gramann or exterior algebra over Cn,R = Cn :=

p

pCn , (4.2a)with the Z2-grading being

R =

p

2pCn

=: R0

p

2p+1Cn

=: R1

. (4.2b)

An R-module M is a Z2-graded bi-module which satisfiesrm = ()prpmmr , (4.3)

24

for homogeneous r R, m M , with M = M0 M1. Then there is a natural map11 , calledthe parity operator, which is defined by

(M)0 := M1 and (M)1 := M0 . (4.4)

We should stress that R is an R-module itself, and as such R is an R-module, as well. However,

R is no longer a Z2-graded ring since (R)1(R)1 (R)1, for instance.A free module of rank m|n over R is defined by

Rm|n := Rm (R)n , (4.5)

where Rm := R R. This has a free system of generators, m of which are bosonic andn of which are fermionic, respectively. We stress that the decomposition of Rm|n into Rm|0 and

R0|n has, in general, no invariant meaning and does not coincide with the decomposition into

bosonic and fermionic parts, [Rm0 (R1)n] [Rm1 (R0)n]. Only when R1 = 0, are thesedecompositions the same. An example is Cm|n, where we consider the complex numbers as aZ2-graded ring (where R = R0 with R0 = C and R1 = 0).

Let R be a supercommutative ring and Rm|n be a freely generated R-module. Just as in the

commutative case, morphisms between free R-modules can be given by matrices. The standard

matrix format is

A =

(

A1 A2

A3 A4

)

, (4.6)

where A is said to be bosonic (respectively, fermionic) if A1 and A4 are filled with bosonic (re-

spectively, fermionic) elements of the ring while A2 and A3 are filled with fermionic (respectively,

bosonic) elements. Furthermore, A1 is a pm-, A2 a qm-, A3 a pn- and A4 a qn-matrix.The set of matrices in standard format with elements in R is denoted by Mat(m|n, p|q,R). Itforms a Z2-graded module which, with the usual matrix multiplication, is naturally isomorphicto Hom(Rm|n, Rp|q). We denote the endomorphisms of Rm|n by End(m|n,R) and the automorph-isms by Aut(m|n,R), respectively. We use further the special symbols gl(m|n,R) End(m|n,R)to denote the bosonic endomorphisms of Rm|n and GL(m|n,R) Aut(m|n,R) to denote thebosonic automorphisms.

The supertranspose of A Mat(m|n, p|q,R) is defined according to

Ast :=

(

At1 ()pA At3()pAAt2 At4

)

, (4.7)

where the superscript t denotes the usual transpose. The supertransposition satisfies (A+B)st =

Ast+Bst and (AB)st = ()pApBBstAst. We shall use the following definition of the supertrace ofA End(m|n,R):

strA := trA1 ()pAtrA4 . (4.8)11More precisely, it is a functor from the category of R-modules to the category of R-modules. See Appendix

C. for details.

25

The supercommutator for matrices is defined analogously to (4.1), i.e. [A,B} := AB()pApBBAfor A,B End(m|n,R). Then str[A,B} = 0 and strAst = strA. Finally, let A GL(m|n,R).The superdeterminant is given by

sdetA := det(A1 A2A14 A3) detA14 , (4.9)

where the right-hand side is well-defined for A1 GL(m|0, R0) and A4 GL(n|0, R0). Further-more, it belongs to GL(1|0, R0). The superdeterminant satisfies the usual rules, sdet(AB) =sdetA sdetB and sdetAst = sdetA for A,B GL(m|n,R). Notice that sometimes sdet is referredto as the Berezinian and also denoted by Ber.

After this digression, we may now introduce the local model of a supermanifold. Let V be an

open subset in Cm and consider OV (Cn) := OV Cn , where OV is the sheaf of holomorphicfunctions on V Cm which is also referred to as the structure sheaf of V . Thus, OV (Cn) is asheaf of supercommutative rings consisting of Cn-valued holomorphic functions on V . Let now(x1, . . . , xm) be coordinates on V Cm and (1, . . . , n) be a basis of the sections of Cn = 1Cn.Then (x1, . . . , xm, 1, . . . , n) are coordinates for the ringed space V

m|n := (V,OV (Cn)). Anyfunction f can thus be Taylor-expanded as

f(x, ) =

I

IfI(x) , (4.10)

where I is a multi-index. These are the fundamental functions in supergeometry.

To define a general supermanifold, let X be some topological space of real dimension 2m, and

let RX be a sheaf of supercommutative rings on X. Furthermore, let N be the ideal subsheafin RX of all nil-potent elements in RX , and define OX := RX/N .12 Then Xm|n := (X,RX ) iscalled a complex supermanifold of dimension m|n if the following is fulfilled:

(i) Xm := (X,OX ) is an m-dimensional complex manifold which we call the body of Xm|n.(ii) For each point x X there is a neighbourhood U x such that there is a local isomorphism

RX |U = OX((N/N 2))|U , whereN/N 2 is a rank-n locally free sheaf ofOX -modules onX,i.e. N/N 2 is locally of the form OX OX (n-times); N/N 2 is called the characteristicsheaf of Xm|n.

Therefore, complex supermanifolds look locally like V m|n = (V,OV (Cn)). In view of this,we picture Cm|n as (Cm,OCm(Cn)). We shall refer to RX as the structure sheaf of thesupermanifold Xm|n and to OX as the structure sheaf of the body Xm of Xm|n. Later on, weshall use a more common notation and re-denoteRX by OX or simply by O if there is no confusionwith the structure sheaf of the body Xm of Xm|n. In addition, we sometimes write Xm|0 instead

of Xm. Furthermore, the tangent bundle TXm|n of a complex supermanifold Xm|n is an example

of a supervector bundle, where the transition functions are sections of the (non-Abelian) sheaf

GL(m|n,RX) (see Section 5.2. for more details).

12Instead of RX , one often also writes R and likewise for OX .

26

Remark 4.1. Recall that for a ringed space (X,OX ) with the property that for each x Xthere is a neighbourhood U x such that there is a ringed space isomorphism (U,OX |U ) =(V,OV ), where V Cm. Then X can be given the structure of a complex manifold andmoreover, any complex manifold arises in this manner. By the usual abuse of notation,

(X,OX ) is often denoted by X.

An important example of a supermanifold in the context of twistor geometry is the complex

projective superspace CPm|n. It is given byCPm|n := (CPm,OCPm((O(1) Cn))) , (4.11)where O(1) is the tautological line bundle over the complex projective space CPm. It is definedanalogously to CP 1 (see (1.19)). The reason for the appearance of O(1) is as follows. If we let(z0, . . . , zm, 1, . . . , n) be homogeneous coordinates

13 on CPm|n, a holomorphic function f onCPm|n has the expansionf =

i1 irf i1ir(z0, . . . , zm) . (4.12)

Surely, for f to be well-defined the homogeneity of f must be zero. Hence, f i1ir = f [i1ir ]

must be of homogeneity r. This explains the above form of the structure sheaf of the complexprojective superspace.

Exercise 4.1. Let E X be a holomorphic vector bundle over a complex manifold X.Show that (X,OX (E)) is a supermanifold according to our definition given above.

Supermanifolds of the form as in the above exercise are called globally split. We see thatCPm|n is of the type E CPm with E = O(1)Cn. Due to a theorem of Batchelor [21] (see alsoe.g. [19]), any smooth supermanifold is globally split. This is due to the existence of a (smooth)

partition of unity. The reader should be warned that, in general, complex supermanifolds are not

of this type (basically because of the lack of a holomorphic partition of unity).

4.2. Supertwistor space

Now we have all the necessary ingredients to generalise (1.15) to the supersymmetric setting.

Supertwistors were first introduced by Ferber [22].

Consider M4|2N = C4|2N together with the identificationTM4|2N = H S (4.13)

where the fibres Hx of H over x M4|2N are C2|N and S is again the dotted spin bundle. Inthis sense, H is of rank 2|N and H = E S, where S is the undotted spin bundle and E is the

13Note that they are subject to the identification (z0, . . . , zm, 1, . . . , n) (tz0, . . . , tzm, t1, . . . , tn), where

t C \ {0}.27

rank-0|N R-symmetry bundle. In analogy to x x , we now have xM xA = (x, i )for A = (, i), B = (, j), . . . and i, j, . . . = 1, . . . ,N . Notice that the above factorisation of thetangent bundle can be understood as a generalisation of a conformal structure (see Remark 1.1.)

known as para-conformal structure (see e.g. [23]).

As in the bosonic setting, we may consider the projectivisation of S to define the correspond-

ence space F 5|2N := P(S) = C4|2N CP 1. Furthermore, we consider the vector fieldsVA =

A =

xA. (4.14)

They define an integrable rank-2|N distribution on the correspondence space. The resultingquotient will be supertwistor P 3|N :

P 3|N M4|2N

F 5|2N1 2

@@R

(4.15)

The projection 2 is the trivial projection and 1 : (xA, ) 7 (zA, ) = (xA, ), where

(zA, ) = (z, i, ) are homogeneous coordinates on P

3|N .

As before, we may cover P 3|N by two coordinate patches, which we (again) denote by U:

U+ : 1 6= 0 and zA+ :=zA

1and + :=

21

,

U : 2 6= 0 and zA :=zA

2and :=

12

.

(4.16)

On U+ U we have zA+ = +zA and + = 1 . This shows that P 3|N can be identified withCP 3|N \CP 1|N . It can also be identified with the total space of the holomorphic fibrationO(1)C2|N CP 1 . (4.17)

Another way of writing this is O(1)C2O(1)CN CP 1, where is the parity map givenin (4.4). In the following, we shall denote the two patches covering the correspondence space

F 5|2N by U. Notice that Remark 2.3. also applies to P 3|N .

Similarly, we may extend the geometric correspondence: A point x M4|2N corresponds toa projective line Lx = 1(

12 (x)) P 3|N , while a point p = (z, ) P 3|N corresponds to a

2|N -plane in superspace-time M4|2N that is parametrised by xA = xA0 + A, where xA0 is aparticular solution to the supersymmetric incidence relation zA = xA.

4.3. Superconformal algebra

Before we move on and talk about supersymmetric extensions of self-dual YangMills theory, let

us digress a little and collect a few facts about the superconformal algebra. The conformal algebra,

conf4, in four dimensions is a real form of the complex Lie algebra sl(4,C). The concrete realform depends on the choice of signature of space-time. For Euclidean signature we have so(1, 5) =su(4) while for Minkowski and Kleinian signatures we have so(2, 4) = su(2, 2) and so(3, 3) =sl(4,R), respectively. Likewise, the N -extended conformal algebrathe superconformal algebra,conf4|Nis a real form of the complex Lie superalgebra sl(4|N ,C) for N < 4 and psl(4|4,C) for

28

N = 4. For a compendium of Lie superalgebras, see e.g. [24]. In particular, for N < 4 we havesu(4|N ), su(2, 2|N ) and sl(4|N ,R) for Euclidean, Minkowski and Kleinian signatures while forN = 4 the superconformal algebras are psu(4|4), psu(2, 2|4) and psl(4|4,R). Notice that for aEuclidean signature, the number N of supersymmetries is restricted to be even.

The generators of conf4|N are

conf4|N = span{P, L ,K

,D,Rij, A |Qi, Qi, Si, Si

}. (4.18)

Here, P represents translations, L (Lorentz) rotations, K special conformal transformations,

D dilatations and Rij the R-symmetry while Qi, Q

i are the Poincare supercharges and S

i,

Si their superconformal partners. Furthermore, A is the axial charge which absent for N = 4.Making use of the identification (1.12), we may also write

conf4|N = span{P , L , L,K

,D,Rij , A |Qi, Qi, Si, Si

}, (4.19)

where the L , L are symmetric in their indices (see also (3.10)). We may also include a central

extension z = span{Z} leading to conf4|N z, i.e. [conf4|N , z} = 0 and [z, z} = 0.The commutation relations for the centrally extended superconformal algebra conf4|N z are

{Qi, Qj} = jiP , {Si, S

j } = ijK ,

{Qi, Sj} = i[jiL

+ 12

ji (D + Z) + 2

Ri

j 12ji (1 4N )A

],

{Qi, Sj } = i[ijL

+ 12

ij(D Z) 2Rj i + 12

ij(1 4N )A

],

[Rij, Sk ] = i2(

jkS

i 1N

jiS

k ) , [Ri

j , Sk] = i2(ki S

j 1N jiS

k) ,

[L, Si ] = i(Si 12Si) , [L, S

i ] = i(

S

i 12

S

i ) ,

[Si, P ] = Qi , [Si , P ] =

Qi ,

[D,Si] = i2Si , [D,Si ] = i2Si ,[A,Si] = i2S

i , [A,Si ] = i2Si ,[Ri

j , Qk] = i2(jkQi 1N

jiQk) , [Ri

j, Qk] =i2(

ki Q

j 1N

jiQ

k) ,

[L , Qi ] = i(

Qi 12Qi) , [L , Qi ] = i(

Q

i 12

Q

i) ,

[Qi,K ] = S

i , [Q

i,K

] = Si ,[D,Qi] =

i2Qi , [D,Q

i] =

i2Q

i ,

[A,Qi] = i2Qi , [A,Qi] = i2Qi ,[Ri

j , Rkl] = i2(

liRk

j jkRil) ,[D,P] = iP , [D,K

] = iK ,

[L, P ] = i(

P 12P) , [L, P ] = i(

P 12

P) ,

[L,K ] = i(K 12K) , [L,K ] = i(

K

12K

) ,

[L, L ] = i(

L

L) , [L, L ] = i(L L ) ,

[P,K] = i(L + L +

D) .

(4.20)

29

Notice that for N = 4, the axial charge A decouples, as mentioned above. Notice also that uponchosing a real structure, not all of the above commutation relations are independent of each other.

Some of them will be related via conjugation.

If we let (zA, ) = (z, i, ) be homogeneous coordinates on P

3|N , then conf4|N z can berealised in terms of the following vector fields:

P =

z, K = z

, D = i

2

(

z

z

)

,

L = i

(

z

z 1

2z

z

)

, L = i

(

1

2

)

,

Rij = i

2

(

i

j 1N k

k

)

, A = i2i

i,

Z = i2

(

z

z+

+ i

i

)

,

Qi = ii

z, Qi = i

i, Si = iz

i, Si = ii

.

(4.21)

Using z = , =

and ij =

ij for := /z

, := / and i := /i, one

can straightforwardly check that the above commutations relations are satisfied. Furthermore,

we emphasise that we work non-projectively. Working projectively, the central charge Z is absent

(when acting on holomorphic functions), as is explained in Remark 4.2. The non-projective version

will turn out to be more useful in our discussion of scattering amplitudes.

Remark 4.2. Consider complex projective superspace CPm|n. Then we have the canonicalprojection : Cm+1|0 \ {0} C0|m CPm|n. Let now (za, i) = (z0, . . . , zm, 1, . . . , n)be linear coordinates on Cm+1|n (or equivalently, homogeneous coordinates on CPm|n) fora = 0, . . . ,m and i = 1, . . . , n. Then

(

za

za+ i

i

)

= 0

as follows from a direct calculation in affine coordinates which are defined byCPm|n Ua : za 6= 0 and (za(a), i(a)) := (zaza , iza)for a = 0, . . . ,m and a 6= a, i.e. CPm|n = a Ua.Likewise, we have a realisation of conf4|N z in terms of vector fields on the correspondence

space F 5|2N compatible with the projection 1 : F 5|2N P 3|2N , i.e. the vector fields (4.21)are the push-forward via 1 of the vector fields on F 5|2N . In particular, if we take (xA, ) =

30

(x, i , ) as coordinates on F5|2N , where are homogeneous coordinates on CP 1, we have

P =

x, K = xx

x xi

i

+ x

,

D = i(

x

x+

1

2i

i 1

2

)

,

L = i

(

x

x 1

2x

x

)

,

L = i

(

x

x 1

2x

x

)

i(

i

i 1

2

k

k

)

+ i

(

1

2

)

,

Rij = i

2

(

i

j 1N

k

k

)

, A = i2i

i, Z = i

2

,

Qi = ii

x, Qi = i

i,

Si = ix

i, Si = ii x

x ii j

j

+ ii

.

(4.22)

In order to understand these expressions, let us consider a holomorphic function f on F 5|2N

which descends down to P 3|N . Recall that such a function is of the form f = f(xA, ) =

f(x, i , ) since then VAf = 0. Then

xA

f =

zA

f ,

xA

f =

(

xA

zA

+

zA

)

f .

(4.23)

Next let us exemplify the calculation for the generator L. The rest is left as an exercise. Using

the relations (4.23), we find

[

i(

xA

xA 1

2x

C

xC

)

+ i

(

1

2

)xA

]

f =

= i

(

1

2

)zA

f .

(4.24)

Therefore,

1

[

i(

xA

xA 1

2x

C

xC

)

+ i

(

1

2

)]

=

= i

(

1

2

)

,

(4.25)

what is precisely the relation between the realisations of the L-generator on F 5|2N and P 3|N

as displayed in (4.21) and (4.22).

31

Exercise 4.2. Show that all the generators (4.21) are the push-forward under 1 of the

generators (4.22).

It remains to give the vector field realisation of the superconformal algebra on space-time

M4|2N . This is rather trivial, however, since 2 : F 5|2N M4|2N is the trivial projection. Wefind

P =

x, K = xx

x xi

i

,

D = i(

x

x+

1

2i

i

)

,

L = i

(

x

x 1

2x

x

)

,

L = i

(

x

x 1

2x

x

)

i(

i

i 1

2

k

k

)

,

Rij = i

2

(

i

j 1N

k

k

)

, A = i2i

i,

Qi = ii

x, Qi = i

i,

Si = ix

i, Si = ii x

x ii j

j

.

(4.26)

5. Supersymmetric self-dual YangMills theory and the PenroseWard

transform

5.1. PenroseWard transform

By analogy with self-dual YangMills theory, we may now proceed to construct supersymmetrised

versions of this theory within the twistor framework. The construction is very similar to the

bosonic setting, so we can be rather brief.

Take a holomorphic vector bundle E P 3|N and pull it back to F 5|2N . Note that although werestrict our discussion to ordinary vector bundles, everything goes through for supervector bundles

as well. Then the transition function is constant along 1 : F5|2N P 3|2N , i.e. V A f+ = 0 where

the V A are the restrictions of VA to the patches U with F5|2N = U+U. Under the assumption

that E is holomorphically trivial on any Lx = 1(12 (x)) P 3N , we again split f+ according

to f+ = 1+ and hence

1+ V

A + =

1 V

A on U+ U. Therefore, we may again

introduce a Lie algebra-valued one-form that has components only along 1 : F5|2N P 3|2N :

AA = AA =

1 V

A , (5.1)

where AA is -independent. Thus, we find

A = 0 , with A := A +AA (5.2)

32

together with the compatibility conditions,

[A,B}+ [A,B} = 0 . (5.3)

These equations are known as the constraint equations of N -extended supersymmetric self-dualYangMills theory (see e.g. [25, 26]).

Let us analyse these equations a bit more for N = 4. Cases with N < 4 can be obtained fromthe N = 4 equations by suitable truncations. We may write the above constraint equations as

[A,B} = FAB , with FAB = ()pApBFBA . (5.4)

We may then parametrise FAB as

FAB = (F , Fi, F

ij) := (f,12i,ij) . (5.5)

Furthermore, upon using Bianchi identities

[A,B},C}+ ()pA(pB+pC)[B ,C},A}+ ()pC(pA+pB)[C ,A},B} = 0 ,

(5.6)

we find two additional fields,

i := 23

jij and G :=

122i(i) , (5.7)

where we have introduced the common abbreviation ij :=12!ijkl

kl and parentheses mean

normalised symmetrisation. Altogether, we have obtained the fields displayed in Table 5.1. Note

that all these fields are superfields, i.e. they live on M4|8 = C4|8.field f

i

ij i G

helicity 1 12 0 12 1multiplicity 1 4 6 4 1

Table 5.1: Field content of N = 4 supersymmetry self-dual YangMills theory.

The question is, how can we construct fields and their corresponding equations of motion on

M4, since that is what we are actually after. The key idea is to impose the so-called transversal

gauge condition [2729]:

i Ai = 0 . (5.8)

This reduces supergauge transformations to ordinary ones as follows. Generic infinitesimal su-

pergauge transformations are of the form AA = A = A+ [AA, ], where is a bosonicLie algebra-valued function on M4|8. Residual gauge transformations that preserve (5.8) are then

given by

i Ai = 0 = i i = 0 = (x) , (5.9)

33

i.e. we are left with gauge transformations on space-time M4. Then, by defining the recursion

operator D := i i = i i and by using the Bianchi identities (5.6), after a somewhat lengthycalculation we obtain the following set of recursion relations:

DA = 12i

i ,

(1 + D)Ai = j

ij ,

Dij =2[ij] ,

Di =2j ij ,

Di = 12i G +

12

j [

jk, ki] ,

DG =2i ([j),

ij ] ,

(5.10)

where, as before, parentheses mean normalised symmetrisation while the brackets denote norm-

alised anti-symmetrisation of the enclosed indices. These equations determine all superfields to

order n+ 1, provided one knows them to n-th order in the fermionic coordinates.

At this point, it is helpful to present some formul which simplify this argument a great deal.

Consider some generic superfield f . Its explicit -expansion has the form

f =f +

k11j1

kjk

f j1jk1k . (5.11)

Here and in the following, the circle refers to the zeroth-order term in the superfield expansion of

some superfield f . Furthermore, we have Df = 1j1 [ ]j11, where the bracket [ ]j11 is a composite

expression of some superfields. For example, we have DA =121j1 []

j11, with []j11 =

1j1 . Now letD [ ]j1jk1k =

k+1jk+1

[ ]j1jk+11k+1 . (5.12)

Then we find after a successive application of D

f =f +

k1

1

k!1j1

kjk

[ ]j1jk1k . (5.13)

If the recursion relation of f is of the form (1 + D)f = 1j1 [ ]j11

as it happens to be for Ai, thenf = 0 and the superfield expansion is of the form

f =

k1

k

(k + 1)!1j1

kjk

[ ]j1jk1k . (5.14)

Using these expressions, one obtains the following results for the superfields A and Ai:

A =A +

12

i

i + ,

Ai = 12! ijj

2

3! ijkl

k

l

j +

+ 324!ijkl(

G

ml + [

mn,

nl])

k

m

j + .

(5.15)

34

Upon substituting these superfield expansions into the constraint equations (5.3), (5.4), we

obtain

f = 0 ,

i = 0 ,

ij = 12

{i,j} ,

i = [

ij,

j] ,

G = {

i,

i}+ 12 [

ij ,

ij] .

(5.16)

These are the equations of motion of N = 4 supersymmetric self-dual YangMills theory. Theequations for less supersymmetry are obtained from these by suitable truncations. We have also

introduced the abbreviation := 12

. We stress that (5.16) represent the field

equations to lowest order in the superfield expansions. With the help of the recursion operator

D , one may verify that they are in one-to-one correspondence with the constraint equations (5.3).

For details, see e.g. [2729].

Altogether, we have a supersymmetric extension of Wards theorem 3.1.:

Theorem 5.1. There is a one-to-one correspondence between gauge equivalence classes of solu-

tions to the N -extended supersymmetric self-dual YangMills equations on space-time M4 andequivalence classes of holomorphic vector bundles over supertwistor space P 3|N which are holo-

morphically trivial on any projective line Lx = 1(12 (x)) P 3|N .

Exercise 5.1. Verify all equations from (5.10) to (5.16).

Finally, let us emphasise that the field equations (5.16) also follow from an action principle.

Indeed, upon varying

S =

d4x tr

{ G

f +

i

i 12

ij

ij + 12

ij{

i,

j}

}

, (5.17)

we find (5.16). In writing this, we have implicitly assumed that a reality condition corresponding

either to Euclidean or Kleinian signature has been chosen; see below for more details. This action

functional is known as the Siegel action [30].

35

Remark 5.1. Let us briefly comment on hidden symmetry structurs of self-dual YangMills

theories. Since Pohlmeyers work [31], it has been known that self-dual YangMills theory

possess infinitely many hidden non-local symmetries. Such symmetries are accompanied

by conserved non-local charges. As was shown in [3236, 14], these symmetries are affine

extensions of internal symmetries with an underlying KacMoody structure. See [37] for

a review. Subsequently, Popov & Preitschopf [38] found affine extensions of conformal

symmetries of KacMoody/Virasoro-type. A systematic investigation of symmetries based

on twistor and cohomology theory was performed in [39] (see also [40,41] and the text book

[1]), where all symmetries of the self-dual YangMills equations were derived. In [42, 43]

(see [44] for a review), these ideas were extended to N -extended self-dual YangMills theory.For some extensions to the full N = 4 supersymmetric YangMills theory, see [45]. Noticethat the symmetries of the self-dual YangMills equations are intimitately connected with

one-loop maximally-helicity-violating scattering amplitudes [4649]. See also Part II of

these lecture notes.

Exercise 5.2. Verify that the action functional (5.17) is invariant under the following

supersymmetry transformations (i is some constant anti-commuting spinor):

A = 12

i

i ,

ij =

2[i

j] ,

i =

2j

ij ,

i = 12

i

G +

12

j [

jk,

ki] ,

G =

2i ([

j),

ij ] .

5.2. Holomorphic ChernSimons theory

Let us pause for a moment and summarise what we have achieved so far. In the preceding sections,

we have discussed N = 4 supersymmetric self-dual YangMills theory by means of holomorphicvector bundles E P 3|4 over the supertwistor space P 3|4 that are holomorphically trivial onall projective lines Lx = 1(

12 (x)) P 3|4. These bundles are given by holomorphic transition

functions f = {f+}. We have further shown that the field equations of N = 4 supersymmetricself-dual YangMills theory arise upon varying a certain action functional on space-time, the

Siegel action. Figure 5.1. summarises pictorially our previous discussion.

The question that now arises and which is depicted in Figure 5.1. concerns the formulation

of a corresponding action principle on the supertwistor space. Certainly, such an action, if it

36

exists, should correspond to the Siegel action on space-time. However, in constructing such

a twistor space action, we immediately face a difficulty. Our above approach to the twistor

re-formulation of field theories, either linear or non-linear, is intrinsically on-shell: Holomorphic

functions on twistor space correspond to solutions to field equations on space-time and vice versa.

In particular, holomorphic transition functions of certain holomorphic vector bundles E P 3|4

correspond to solutions to the N = 4 supersymmetric self-dual YangMills equations. Therefore,we somehow need an off-shell approach to holomorphic vector bundles, that is, we need a theory

on supertwistor space that describes complex vector bundles such that the on-shell condition is

the holomorphicity of these bundles.

?

Siegel action

of N = 4 supersymmetricself-dual YangMills theory

holomorphic vector bundles

E P 3|4

trivial on any Lx P 3|4

given by f = {f+}

solutions to the N = 4supersymmetric self-dual

YangMills equations

on M4

66

-

-

Figure 5.1: Correspondences between supertwistor space and space-time.

Before we delve into this issue, let us formalise our above approach to holomorphic vector

bundles (which is also known as the Cech approach). Consider a complex (super)manifold (X,O)with an open covering U = {Ui}. We are interested in holomorphic maps from open subsets ofX into GL(r,C) as well as in the sheaf GL(r,O) of such matrix-valued functions.14 Notice thatGL(r,O) is a non-Abelian sheaf contrary to the Abelian sheaves considered so far. A q-cochainof the covering U with values in GL(r,O) is a collection f = {fi0iq} of sections of the sheafGL(r,O) over non-empty intersections Ui0 Uiq . We will denote the set of such q-cochainsby Cq(U,GL(r,O)). We stress that it has a group structure, where the multiplication is justpointwise multiplication.

We may define the subsets of cocycles Zq(U,GL(r,O)) Cq(U,GL(r,O)). For example, forq = 0, 1 they are given by

Z0(U,GL(r,O)) := {f C0(U,GL(r,O)) | fi = fj on Ui Uj 6= } ,Z1(U,GL(r,O)) := {f C1(U,GL(r,O)) | fij = f1ji on Ui Uj 6=

and fijfjkfki = 1 on Ui Uj Uk 6= } .

(5.18)

14Basically everything we shall say below will also apply to GL(r|s,O) and hence to supervector bundles. As we

are only concerned with ordinary vector bundles (after all we are interested in SU(r) gauge theory), we will stick

to GL(r,O) for concreteness. See e.g. [44] for the following treatment in the context of supervector bundles.

37

These sets will be of particular interest. We remark that from the first of these two definitions

it follows that Z0(U,GL(r,O)) coincides with the group of global sections, H0(U,GL(r,O)), ofthe sheaf GL(r,O). Note that in general the subset Z1(U,GL(r,O)) C1(U,GL(r,O)) is nota subgroup of the group C1(U,GL(r,O)). For notational reasons, we shall denote elements ofC0(U,GL(r,O)) also by h = {hi}.

We say that two cocycles f, f Z1(U,GL(r,O)) are equivalent if f ij = h1i fijhj for someh C0(U,GL(r,O)), since one can always absorb the h = {hi} in a re-definition of the framefields. Notice that this is precisely the transformation we already encountered in (3.23). The set

of equivalence classes induced by this equivalence relation is the first Cech cohomology set and de-

noted by H1(U,GL(r,O)). If the Ui are all Stein (see Remark 5.2.)in the case of supermanifoldsX we need the body to be covered by Stein manifoldswe have the bijection

H1(U,GL(r,O)) = H1(X,GL(r,O)) , (5.19)

otherwise one takes the inductive limit (see Remark 2.2.).

Remark 5.2. We call an ordinary complex manifold (X,O) Stein if X is holomorphicallyconvex (that is, the holomorphically convex hull of any compact subset of X is again compact

in X) and for any x, y X with x 6= y there is some f O such that f(x) 6= f(y).

To sum up, we see that within the Cech approach, rank-r holomorphic vector bundles over some

complex (super)manifold X are parametrised by H1(X,GL(r,O)). Notice that our cover {U} ofthe (super)twistor space is Stein and so H1({U},GL(r,O)) = H1(P 3|N ,GL(r,O)). This in turnexplains that all of our above constructions are independent of the choice of cover.

Another approach to holomorphic vector bundles is the so-called Dolbeault approach. Let X

be a complex (super)manifold and consider a rank-r complex vector bundleE X. Furthermore,we let p,q(X) be the smooth differential (p, q)-forms on X and : p,q(X) p,q+1(X) be theanti-holomorphic exterior derivative. A (0, 1)-connection on E is defined by a covariant differential

0,1 : p,q(X,E) p,q+1(X,E) which satisfies the Leibniz formula. Here, p,q(X,E) :=p,q(X) E. Locally, it is of the form 0,1 = + A0,1, where A0,1 is a differential (0, 1)-formwith values in EndE which we shall refer to as the connection (0, 1)-form. The complex vector

bundle E is said to be holomorphic if the (0, 1)-connection is flat, that is, if the corresponding

curvature

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